Edit report at http://bugs.php.net/bug.php?id=53918&edit=1
ID: 53918
Updated by: cataphr...@php.net
Reported by: exploringbinary at gmail dot com
Summary: printf of floating point variable prints maximum of
53 decimal places
Status: Open
Type: Bug
Package: Math related
Operating System: Windows (32-bit system)
PHP Version: 5.3.5
Block user comment: N
Private report: N
New Comment:
I'm sorry, my classes of numerical analysis are long gone.
First let's establish what we're talking about. Let's take Mathematica's
definitions:
http://reference.wolfram.com/mathematica/ref/Precision.html
Precision[x] gives the effective number of digits of precision in the
number x.
Precision[x] gives a measure of the relative uncertainty in the value of
x.
With absolute uncertainty dx, Precision[x] is -Log[10, dx/x].
The meaning of "absolute uncertainty" gets clear by this passage:
«Mathematica is set up so that if a number x has uncertainty dx, then
its true value can lie anywhere in an interval of size dx from x-dx/2 to
x+dx/2.»
So, in this convention, 60.0000000000000000 (60. + 16 zeros) represents
60 ± 0.5e-16 -- 60 with an absolute uncertainty of 1e-16. We need a
precision greater than double precision, of -log10(1e-16/60), or
17.7782.
The (implicit) *precision* of a (normalized) double is _always_
-log10(2^-53), or around 15.9546. These are the 52 bits in the mantissa
plus an implicit bit in the normalized doubles.
Now for accuracy:
http://reference.wolfram.com/mathematica/ref/Accuracy.html
Accuracy[x] gives the effective number of digits to the right of the
decimal point in the number x.
Accuracy[x] gives a measure of the absolute uncertainty in the value of
x.
With uncertainty dx, Accuracy[x] is -Log[10,dx].
Accuracy[x] is equal to Precision[x]-RealExponent[x] (where
RealExponent[x] is log10(abs(x)))
>From these definitions, it becomes clear that what you want matters for
this bug report is the *accuracy* in the representation, i.e., only the
digits to the right of the decimal point matter.
It's also clear that the smaller the (absolute value of) number, greater
is the accuracy. In fact, by the relationship given above, the accuracy
of a double is given by:
15.9546 - log10(abs(x))
And the accuracy needed to represent the effective digits of the
smallest normalized double, 2.2250738585072014e-308 (2^-1022), is
323.607. You can have smaller denormalized numbers (for which
-RealExponent gets larger), but the precision also dwindles. The
smallest possible denormalized double, 2^-1074, only has one effective
"binary digit" of precision. The precision is log10(2), or 0.30103;
adding -RealExponent(2^-1074) gives the same accuracy of 323.607.
So would a limit of 324 digits be enough? Yes, but only if you want to
show only the effective digits. If you want to show the exact number
represented by the double, you'll need a lot more digits.
For instance, to fully represent the value stored in the double
001ffffffffffffe, you'd need 766 decimal digits, not counting the 307
leading zeros. See
http://www.wolframalpha.com/input/?i=2%5E-1074%2A16%5E%5E1FFFFFFFFFFFFE
In my opinion, it's deceiving to show decimal places beyond the
effective digits. Those numbers are rendered useless by the rounding
errors. Except for some exotic application that would that uses numbers
only in the domain of the numbers representable in the IEEE double, I
fail to see any point in showing them.
The current limit of 53 decimal places, albeit arbitrary, is more than
generous. 16 decimal places would be enough to show the effective digits
in scientific notation. The remaining leave some space to show leading
zeros in non-scientific representations and insignificant digits.
Then there's the problem that not all C libraries support these
accuracies.
Therefore, I would mark both this bug and bug #47168 as Wont Fix.
Previous Comments:
------------------------------------------------------------------------
[2011-02-03 15:19:05] exploringbinary at gmail dot com
I got the same "this is not a bug" response for bug #47168 -- until
Rasmus stepped in that is. Please reconsider.
I COULD print the numbers in exponential format, but that's besides the
point. If you accept that bug #47168 was valid, then this bug should be
too.
The "15.9546 decimal digits" of precision does not apply here; my
examples are powers of two: every digit is accurate.
------------------------------------------------------------------------
[2011-02-03 14:28:35] cataphr...@php.net
Well, can't you just print the numbers in exponential format?
A representation with more than 50 leading digits in the fractional part
is not particularly useful.
This seems a better option than showing > 53 digits of decimal places
when the machine precision is of only 15.9546 decimal digits.
I imagine the magic arbitrary 53 digits limits was because the precision
of the machine precision of 53 binary digits, though this is of course
irrelevant for decimal representations.
------------------------------------------------------------------------
[2011-02-03 14:23:24] il...@php.net
Thank you for taking the time to write to us, but this is not
a bug. Please double-check the documentation available at
http://www.php.net/manual/ and the instructions on how to report
a bug at http://bugs.php.net/how-to-report.php
53 digits is the maximum limit enforced by most programming languages.
http://www.exploringbinary.com/print-precision-of-dyadic-fractions-varies-by-
language/
------------------------------------------------------------------------
[2011-02-03 12:56:28] exploringbinary at gmail dot com
Description:
------------
Bug #47168, "printf of floating point variable prints maximum of 40
decimal places", was not fixed as expected. Instead of the previous
arbitrary limit of 40 digits, there is now an arbitrary limit of 53
digits.
These three examples all print powers of two, which have exact
representations in double-precision floating-point:
<?php
printf("%1.176f\n",1.044048714879763924273647057481047608912186281291034647641381832875155719135597796355663380296618925058282911777496337890625e-53);
/* 2^-176 */?>
<?php
printf("%1.177f\n",5.220243574398819621368235287405238044560931406455173238206909164375778595677988981778316901483094625291414558887481689453125e-54);
/* 2^-177 */?>
<?php
printf("%1.178f\n",2.6101217871994098106841176437026190222804657032275866191034545821878892978389944908891584507415473126457072794437408447265625e-54);
/* 2^-178 */?>
The output is:
0.00000000000000000000000000000000000000000000000000001
0.00000000000000000000000000000000000000000000000000001
0.00000000000000000000000000000000000000000000000000000
(The first prints 1 significant digit, the second rounds to 1
significant digit, and the third prints no significant digits.)
Compare this to gcc C on Linux:
printf("%1.176f\n",1.044048714879763924273647057481047608912186281291034647641381832875155719135597796355663380296618925058282911777496337890625e-53);
/* 2^-176 */
printf("%1.177f\n",5.220243574398819621368235287405238044560931406455173238206909164375778595677988981778316901483094625291414558887481689453125e-54);
/* 2^-177 */
printf("%1.178f\n",2.6101217871994098106841176437026190222804657032275866191034545821878892978389944908891584507415473126457072794437408447265625e-54);
/* 2^-178 */
The output (which is correct) is:
0.00000000000000000000000000000000000000000000000000001044048714879763924273647057481047608912186281291034647641381832875155719135597796355663380296618925058282911777496337890625
0.000000000000000000000000000000000000000000000000000005220243574398819621368235287405238044560931406455173238206909164375778595677988981778316901483094625291414558887481689453125
0.0000000000000000000000000000000000000000000000000000026101217871994098106841176437026190222804657032275866191034545821878892978389944908891584507415473126457072794437408447265625
(I see that in formatted_print.c,
#define MAX_FLOAT_PRECISION 40
was changed to
#define MAX_FLOAT_PRECISION 53)
Test script:
---------------
<?php
printf("%1.178f\n",2.6101217871994098106841176437026190222804657032275866191034545821878892978389944908891584507415473126457072794437408447265625e-54);
/* 2^-178 */?>
Expected result:
----------------
All requested decimal places are printed.
Actual result:
--------------
Only 53 decimal places are printed.
------------------------------------------------------------------------
--
Edit this bug report at http://bugs.php.net/bug.php?id=53918&edit=1
